Abstract: 
The space of the closures of leaves of a Riemannian foliation is a
nice topological space, a stratified singular space which can be
topologically embedded in $\mathbb{R}^k$ for $k$ sufficiently large. In
the case of Orbit Like Foliations (OLF) the smooth structure
induced by the embedding and the smooth structure defined by basic
functions is the same. We study geometric structures adapted to
the foliation and present conditions which assure that the given
structure descends to the leaf closure space. In Section 5 we
introduce the notion of an Ehresmann connection on a stratified
foliated space and study the properties of the strata which depend
on the existence of such a connection. We also give conditions
which ensure that a connection understood as a differential
operator defines an Ehresmann connection as above. In the last
section we present some curvature estimates for metric structures
on the leaf closure space.

1. Instytut Matematyki
   Uniwersytet Jagielloñski
   Reymonta 4
   30-059 Kraków, Poland
   
2. Instytut Matematyki
   Uniwersytet Jagielloñski
   Reymonta 4
   30-059 Kraków, Poland